# Column vector notation as “ordered set notation”

AFAIK, a vector can be specified using either "ordered set notation" or "matrix notation"

Ordered set notation

Matrix notation of row and colum vectors

I wonder if a column vector can be specified using ordered set notation. For example, can a column vector

$$\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$$ be specified as follows? $$(1,2,3)$$

Plus, is the following statement correct?

A set {(1,0,0), (1,1,0), (1,0,1)} is a basis of column space of the matrix

$$\begin{bmatrix} 1&2&1&1\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}$$.

• Yep. Why do you think not? – Aniruddh Agarwal Apr 29 '20 at 3:20
• Some write $(1\;2 \;3)^T$ for $\pmatrix{1\\2\\3}$ – J. W. Tanner Apr 29 '20 at 3:23
• In the context of vectors & matrices, I always interpret $(1,2,3)$ as a column vector. It's a way to write a column vector without leaving lots of white space on the page. – Gerry Myerson Apr 29 '20 at 3:27
• My TA said I should've specified the basis vectors as $${{(1,0,0)^T,(1,1,0)^T,(1,0,1)^T}}$$, which I think is non-sense because it is a mix of two notation methods. – hskim Apr 29 '20 at 3:29
• @Gerry_Myerson But how do you write row vectors, then? – J.-E. Pin Apr 29 '20 at 6:50

I think that the column vector $$\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$$ and the ordered set $$(1,2,3)$$ represent the same element of the same linear space.